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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Kugeltensor</span></h1>
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<p><b>Kugeltensoren</b>, <b>Axiatoren</b> oder <b>sphärische Tensoren</b> sind in der <a href="Kontinuumsmechanik" title="Kontinuumsmechanik">Kontinuumsmechanik</a> <a href="Tensor" title="Tensor">Tensoren</a>, die proportional zum <a href="Einheitstensor" title="Einheitstensor">Einheitstensor</a> zweiter Stufe sind; sie sind daher geeignet, Vektoren wie in Abb. 1 dargestellt <a href="Zentrische_Streckung" title="Zentrische Streckung">zentrisch zu strecken</a>. Der Kugel- oder sphärische Anteil eines Tensors <b>T</b> ist der Kugeltensor sph(<b>T</b>)&nbsp;=&nbsp;<b>T</b><sup>K</sup>, der dieselbe <a href="Spur_(Mathematik)" title="Spur (Mathematik)">Spur</a> wie der Tensor <b>T</b> besitzt.
</p><p>Kugeltensoren treten in der Kontinuumsmechanik bei allseitigem, hydrostatischem Druck oder bei in allen drei Raumrichtungen gleichförmiger Expansion oder Kompression eines Körpers auf. Sie werden daher zur Modellierung des Materialverhaltens unter diesen Bedingungen benutzt.
</p>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>Kugeltensoren sind Tensoren zweiter Stufe <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {T} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<mi mathvariant="bold">T</mi>
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<annotation encoding="application/x-tex">{\displaystyle \mathbf {T} }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9593e3b995a1b57c078873a5ea186c7012e1a5ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.859ex; height:2.176ex;" alt="{\displaystyle \mathbf {T} }" loading="lazy"></span>, die das <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \in \mathbb {R} }">
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<annotation encoding="application/x-tex">{\displaystyle \lambda \in \mathbb {R} }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b87bc1622689bc998795834cd65eecdb4955a785.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.874ex; height:2.176ex;" alt="{\displaystyle \lambda \in \mathbb {R} }" loading="lazy"></span> fache des <a href="Einheitstensor" title="Einheitstensor">Einheitstensors</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1} }">
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</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {T} :\quad \mathbf {T} =\lambda \mathbf {1} }">
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<annotation encoding="application/x-tex">{\displaystyle \mathbf {T} :\quad \mathbf {T} =\lambda \mathbf {1} }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48e36efb2260ff3a1e9e0995d7de001b32595972.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.769ex; height:2.176ex;" alt="{\displaystyle \mathbf {T} :\quad \mathbf {T} =\lambda \mathbf {1} }" loading="lazy"></span>.</dd></dl>
<p>Der Kugelanteil eines Tensors <b>T</b> wird mit einem hochgestellten "K" oder "sph" bezeichnet:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {T} ^{\mathrm {K} }=\mathrm {sph} (\mathbf {T} ):={\frac {\mathrm {Sp} (\mathbf {T} )}{\operatorname {Sp} (\mathbf {1} )}}\mathbf {1} ={\frac {\mathrm {Sp} (\mathbf {T} )}{3}}\mathbf {1} }">
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<annotation encoding="application/x-tex">{\displaystyle \mathbf {T} ^{\mathrm {K} }=\mathrm {sph} (\mathbf {T} ):={\frac {\mathrm {Sp} (\mathbf {T} )}{\operatorname {Sp} (\mathbf {1} )}}\mathbf {1} ={\frac {\mathrm {Sp} (\mathbf {T} )}{3}}\mathbf {1} }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/243a6569d77dadd564434314a2a2066f796dbe99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:37.335ex; height:6.509ex;" alt="{\displaystyle \mathbf {T} ^{\mathrm {K} }=\mathrm {sph} (\mathbf {T} ):={\frac {\mathrm {Sp} (\mathbf {T} )}{\operatorname {Sp} (\mathbf {1} )}}\mathbf {1} ={\frac {\mathrm {Sp} (\mathbf {T} )}{3}}\mathbf {1} }" loading="lazy"></span>.</dd></dl>
<p>Die <a href="Spur_(Mathematik)" title="Spur (Mathematik)">Spur</a> "Sp" des <a href="Einheitstensor" title="Einheitstensor">Einheitstensors</a> <b>1</b> ist gleich der Dimension des zugrunde gelegten Raumes, hier und im Folgenden gleich drei.
</p>
<div class="mw-heading mw-heading2"><h2 id="Expansion_und_Kompression">Expansion und Kompression</h2></div>

<p>Wie eingangs erwähnt treten Kugeltensoren bei in allen drei Raumrichtungen gleichförmiger Expansion oder Kompression eines Körpers auf, die wie folgt beschrieben werden kann. In der Kontinuumsmechanik gibt die Bewegungsfunktion <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}={\vec {\chi }}({\vec {X}},t)}">
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e92bff89d59a995104a9f1d246741c880d1b2b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.546ex; height:3.343ex;" alt="{\displaystyle {\vec {x}}={\vec {\chi }}({\vec {X}},t)}" loading="lazy"></span> den Ort <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}}">
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/db2dc6ced9cc3bc7e8b9f2707cbec033f6d3759c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {x}}}" loading="lazy"></span> an, an dem zur Zeit <i>t</i> ein Partikel ist, das zu einer definierten Zeit <i>t</i><sub>0</sub> am Ort <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {X}}}">
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5dc35b5a0226cf11a2c3f2d2dbbac6ab5ade6036.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.843ex;" alt="{\displaystyle {\vec {X}}}" loading="lazy"></span> war. Die Zahlen <i>X</i><sub>1,2,3</sub>&nbsp;∈&nbsp;ℝ sind die Koordinaten des Vektors <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {X}}}">
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</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}={\vec {\chi }}({\vec {X}},t)=\lambda \left({\vec {X}}-{\vec {o}}\right)+{\vec {o}}}">
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<annotation encoding="application/x-tex">{\displaystyle {\vec {x}}={\vec {\chi }}({\vec {X}},t)=\lambda \left({\vec {X}}-{\vec {o}}\right)+{\vec {o}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f702adc067be1d36c3a4f0d9afec29046505d15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.303ex; height:4.843ex;" alt="{\displaystyle {\vec {x}}={\vec {\chi }}({\vec {X}},t)=\lambda \left({\vec {X}}-{\vec {o}}\right)+{\vec {o}}}" loading="lazy"></span></dd></dl>
<p>für alle Partikel gilt, siehe Abb. 2. Bildung des <a href="Gradient_eines_Vektorfeldes" class="mw-redirect" title="Gradient eines Vektorfeldes">Gradienten</a> nach den <i>materiellen</i> Koordinaten <i>X</i><sub>1,2,3</sub> liefert den <a href="Deformationsgradient" title="Deformationsgradient">Deformationsgradient</a>
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {F} :=&amp;\operatorname {GRAD} ({\vec {\chi }}):=\sum _{i,j=1}^{3}{\frac {\mathrm {d} \chi _{i}}{{\mathrm {d} X}_{j}}}{\vec {e}}_{i}\otimes {\vec {e}}_{j}=\sum _{i,j=1}^{3}{\frac {\mathrm {d} }{\mathrm {d} X_{j}}}[\lambda (X_{i}-o_{i})+o_{i}]{\vec {e}}_{i}\otimes {\vec {e}}_{j}\\=&amp;\sum _{i,j=1}^{3}\lambda \delta _{ij}{\vec {e}}_{i}\otimes {\vec {e}}_{j}=\lambda \mathbf {1} \end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<mtr>
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<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">F</mi>
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<mo>:=</mo>
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<mi>GRAD</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
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<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>,</mo>
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<mi>χ<!-- χ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
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</msub>
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<msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
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<mi>X</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
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<mo>=</mo>
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<mo>∑<!-- ∑ --></mo>
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<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
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<mi mathvariant="normal">d</mi>
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<mi>X</mi>
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<mi>j</mi>
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<mo stretchy="false">[</mo>
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<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
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<mi>o</mi>
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<mi>i</mi>
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<mo stretchy="false">]</mo>
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<mi>i</mi>
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<mrow class="MJX-TeXAtom-ORD">
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<mi>j</mi>
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<mo>=</mo>
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<mtd>
<mi></mi>
<munderover>
<mo>∑<!-- ∑ --></mo>
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<mi>λ<!-- λ --></mi>
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<mo>=</mo>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn mathvariant="bold">1</mn>
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {F} :=&amp;\operatorname {GRAD} ({\vec {\chi }}):=\sum _{i,j=1}^{3}{\frac {\mathrm {d} \chi _{i}}{{\mathrm {d} X}_{j}}}{\vec {e}}_{i}\otimes {\vec {e}}_{j}=\sum _{i,j=1}^{3}{\frac {\mathrm {d} }{\mathrm {d} X_{j}}}[\lambda (X_{i}-o_{i})+o_{i}]{\vec {e}}_{i}\otimes {\vec {e}}_{j}\\=&amp;\sum _{i,j=1}^{3}\lambda \delta _{ij}{\vec {e}}_{i}\otimes {\vec {e}}_{j}=\lambda \mathbf {1} \end{aligned}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b60dd009ed53b18f2dd64a5ae8a0965acfe9678.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:71.711ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}\mathbf {F} :=&amp;\operatorname {GRAD} ({\vec {\chi }}):=\sum _{i,j=1}^{3}{\frac {\mathrm {d} \chi _{i}}{{\mathrm {d} X}_{j}}}{\vec {e}}_{i}\otimes {\vec {e}}_{j}=\sum _{i,j=1}^{3}{\frac {\mathrm {d} }{\mathrm {d} X_{j}}}[\lambda (X_{i}-o_{i})+o_{i}]{\vec {e}}_{i}\otimes {\vec {e}}_{j}\\=&amp;\sum _{i,j=1}^{3}\lambda \delta _{ij}{\vec {e}}_{i}\otimes {\vec {e}}_{j}=\lambda \mathbf {1} \end{aligned}}}" loading="lazy"></span></dd></dl>
<p>der hier ein Kugeltensor ist. Das Rechenzeichen ⊗ bildet das <a href="Dyadisches_Produkt" title="Dyadisches Produkt">dyadische Produkt</a> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{ij}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>δ<!-- δ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \delta _{ij}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa75d04c11480d976e1396951e02cbb3c4f71568.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.51ex; height:3.009ex;" alt="{\displaystyle \delta _{ij}}" loading="lazy"></span> bezeichnet das <a href="Kronecker-Delta" title="Kronecker-Delta">Kronecker-Delta</a>. Die <a href="Determinante" title="Determinante">Determinante</a> des Deformationsgradienten ist das Volumenverhältnis vor und nach der Expansion:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {det} (\mathbf {F} )=\lambda ^{3}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>det</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">F</mi>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msup>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {det} (\mathbf {F} )=\lambda ^{3}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/16e896c438ec779db945172c047ebcc4a66e70c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.23ex; height:3.176ex;" alt="{\displaystyle \operatorname {det} (\mathbf {F} )=\lambda ^{3}}" loading="lazy"></span>.</dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Inkompressibilität"><span id="Inkompressibilit.C3.A4t"></span>Inkompressibilität</h2></div>
<div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→&nbsp;</span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="Inkompressibilit%C3%A4t" title="Inkompressibilität">Inkompressibilität</a></i></div>
<p>Für ein <a href="Inkompressibilit%C3%A4t" title="Inkompressibilität">inkompressibles</a> Material ist die im vorigen Abschnitt beschriebene volumenändernde Deformation unmöglich, denn Inkompressibilität zeichnet sich durch ein konstantes Volumenverhältnis von eins aus. Mathematisch wird dies durch die Nebenbedingung
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {det} (\mathbf {F} )\equiv 1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
<mi mathvariant="normal">e</mi>
<mi mathvariant="normal">t</mi>
</mrow>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="bold">F</mi>
</mrow>
<mo stretchy="false">)</mo>
<mo>≡<!-- ≡ --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathrm {det} (\mathbf {F} )\equiv 1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3195dd1ca421ab3cf5114905dfa11b8baaff0acd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.982ex; height:2.843ex;" alt="{\displaystyle \mathrm {det} (\mathbf {F} )\equiv 1}" loading="lazy"></span></dd></dl>
<p>an die Bewegungsfunktion <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {\chi }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>χ<!-- χ --></mi>
<mo stretchy="false">→<!-- → --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\vec {\chi }}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/21b47209887a879af8c407f22b941a5555c2ffc0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.455ex; height:2.676ex;" alt="{\displaystyle {\vec {\chi }}}" loading="lazy"></span> ausgedrückt. Eine solche Nebenbedingung wird mit einem <a href="Lagrangescher_Multiplikator" class="mw-redirect" title="Lagrangescher Multiplikator">Lagrangeschen Multiplikator</a> sichergestellt, der hier dem Druck <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span> im Material entspricht. Die zugehörige Reaktionsspannung ist der <a href="Drucktensor" class="mw-redirect" title="Drucktensor">Drucktensor</a>
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -p\mathbf {1} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>−<!-- − --></mo>
<mi>p</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn mathvariant="bold">1</mn>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle -p\mathbf {1} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8f0f3e389fa5edbd57d82e63add6985dd2b389c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.314ex; height:2.509ex;" alt="{\displaystyle -p\mathbf {1} }" loading="lazy"></span>,</dd></dl>
<p>der ein Kugeltensor ist. Beispiele für diese Beschreibungsweise finden sich in der <a href="Hyperelastizit%C3%A4t" title="Hyperelastizität">Hyperelastizität</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Ort_im_Eigenwertraum">Ort im Eigenwertraum</h2></div>

<p>Als Vielfaches des Einheitstensors hat jeder Kugeltensor drei identische Eigenwerte <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1,2,3}=\lambda }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
<mo>,</mo>
<mn>3</mn>
</mrow>
</msub>
<mo>=</mo>
<mi>λ<!-- λ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lambda _{1,2,3}=\lambda }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d7ab53bbc34af166fe529e524e50f236b29f171a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.422ex; height:2.843ex;" alt="{\displaystyle \lambda _{1,2,3}=\lambda }" loading="lazy"></span> die im Eigenwertraum auf der hydrostatischen Achse <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1}=\lambda _{2}=\lambda _{3}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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</msub>
<mo>=</mo>
<msub>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msub>
<mo>=</mo>
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<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
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</msub>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lambda _{1}=\lambda _{2}=\lambda _{3}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0825add6c66cc060f5a1a330574048fd88501672.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.425ex; height:2.509ex;" alt="{\displaystyle \lambda _{1}=\lambda _{2}=\lambda _{3}}" loading="lazy"></span> liegen, siehe Abbildung rechts. Diese Achse wird, sofern nur <a href="Symmetrische_Matrix" title="Symmetrische Matrix">symmetrische</a> Tensoren betrachtet werden, von den Kugeltensoren gebildet.
</p>
<div class="mw-heading mw-heading2"><h2 id="Invarianten_von_Kugeltensoren">Invarianten von Kugeltensoren</h2></div>
<p>Die drei <a href="Hauptinvariante" title="Hauptinvariante">Hauptinvarianten</a> eines Kugeltensors lauten
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{rclcl}\mathrm {I} _{1}(\lambda \mathbf {1} )&amp;=&amp;\operatorname {Sp} (\lambda \mathbf {1} )&amp;=&amp;3\lambda \\\mathrm {I} _{2}(\lambda \mathbf {1} )&amp;=&amp;{\frac {1}{2}}[\operatorname {Sp} {(\lambda \mathbf {1} )}^{2}-\operatorname {Sp} (\lambda \mathbf {1} \cdot \lambda \mathbf {1} )]&amp;=&amp;3\lambda ^{2}\\\mathrm {I} _{3}(\lambda \mathbf {1} )&amp;=&amp;\operatorname {det} (\lambda \mathbf {1} )&amp;=&amp;\lambda ^{3}\end{array}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right center left center left" rowspacing="4pt" columnspacing="1em">
<mtr>
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<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn mathvariant="bold">1</mn>
</mrow>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mo>=</mo>
</mtd>
<mtd>
<mi>Sp</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn mathvariant="bold">1</mn>
</mrow>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mo>=</mo>
</mtd>
<mtd>
<mn>3</mn>
<mi>λ<!-- λ --></mi>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">I</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn mathvariant="bold">1</mn>
</mrow>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mo>=</mo>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo stretchy="false">[</mo>
<mi>Sp</mi>
<mo>⁡<!-- ⁡ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn mathvariant="bold">1</mn>
</mrow>
<mo stretchy="false">)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mi>Sp</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn mathvariant="bold">1</mn>
</mrow>
<mo>⋅<!-- ⋅ --></mo>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn mathvariant="bold">1</mn>
</mrow>
<mo stretchy="false">)</mo>
<mo stretchy="false">]</mo>
</mtd>
<mtd>
<mo>=</mo>
</mtd>
<mtd>
<mn>3</mn>
<msup>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">I</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo stretchy="false">(</mo>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn mathvariant="bold">1</mn>
</mrow>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mo>=</mo>
</mtd>
<mtd>
<mi>det</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn mathvariant="bold">1</mn>
</mrow>
<mo stretchy="false">)</mo>
</mtd>
<mtd>
<mo>=</mo>
</mtd>
<mtd>
<msup>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msup>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{rclcl}\mathrm {I} _{1}(\lambda \mathbf {1} )&amp;=&amp;\operatorname {Sp} (\lambda \mathbf {1} )&amp;=&amp;3\lambda \\\mathrm {I} _{2}(\lambda \mathbf {1} )&amp;=&amp;{\frac {1}{2}}[\operatorname {Sp} {(\lambda \mathbf {1} )}^{2}-\operatorname {Sp} (\lambda \mathbf {1} \cdot \lambda \mathbf {1} )]&amp;=&amp;3\lambda ^{2}\\\mathrm {I} _{3}(\lambda \mathbf {1} )&amp;=&amp;\operatorname {det} (\lambda \mathbf {1} )&amp;=&amp;\lambda ^{3}\end{array}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4539bf664be3be6b93da58d70b217201faacef05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:49.402ex; height:10.509ex;" alt="{\displaystyle {\begin{array}{rclcl}\mathrm {I} _{1}(\lambda \mathbf {1} )&amp;=&amp;\operatorname {Sp} (\lambda \mathbf {1} )&amp;=&amp;3\lambda \\\mathrm {I} _{2}(\lambda \mathbf {1} )&amp;=&amp;{\frac {1}{2}}[\operatorname {Sp} {(\lambda \mathbf {1} )}^{2}-\operatorname {Sp} (\lambda \mathbf {1} \cdot \lambda \mathbf {1} )]&amp;=&amp;3\lambda ^{2}\\\mathrm {I} _{3}(\lambda \mathbf {1} )&amp;=&amp;\operatorname {det} (\lambda \mathbf {1} )&amp;=&amp;\lambda ^{3}\end{array}}}" loading="lazy"></span></dd></dl>
<p>Der Betrag ist die <a href="Frobeniusnorm" title="Frobeniusnorm">Frobeniusnorm</a>, die sich mit dem <a href="Frobenius-Skalarprodukt" title="Frobenius-Skalarprodukt">Frobenius-Skalarprodukt</a> "<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle :}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>:</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle :}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd064c6ce80ad9a8e53adebb7ad51b7635fceb0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:1.676ex;" alt="{\displaystyle :}" loading="lazy"></span>" zu
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \parallel \lambda \mathbf {1} \parallel ={\sqrt {\lambda \mathbf {1} :\lambda \mathbf {1} }}={\sqrt {\operatorname {Sp} (\lambda \mathbf {1} \cdot \lambda \mathbf {1} )}}={\sqrt {3}}|\lambda |}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">∥<!-- ∥ --></mo>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn mathvariant="bold">1</mn>
</mrow>
<mo>∥<!-- ∥ -->=</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn mathvariant="bold">1</mn>
</mrow>
<mo>:</mo>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn mathvariant="bold">1</mn>
</mrow>
</msqrt>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mi>Sp</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn mathvariant="bold">1</mn>
</mrow>
<mo>⋅<!-- ⋅ --></mo>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn mathvariant="bold">1</mn>
</mrow>
<mo stretchy="false">)</mo>
</msqrt>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>λ<!-- λ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \parallel \lambda \mathbf {1} \parallel ={\sqrt {\lambda \mathbf {1} :\lambda \mathbf {1} }}={\sqrt {\operatorname {Sp} (\lambda \mathbf {1} \cdot \lambda \mathbf {1} )}}={\sqrt {3}}|\lambda |}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0b3e7ab7370d6cb72de869233aa6d4d6a8856ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:43.097ex; height:4.843ex;" alt="{\displaystyle \parallel \lambda \mathbf {1} \parallel ={\sqrt {\lambda \mathbf {1} :\lambda \mathbf {1} }}={\sqrt {\operatorname {Sp} (\lambda \mathbf {1} \cdot \lambda \mathbf {1} )}}={\sqrt {3}}|\lambda |}" loading="lazy"></span></dd></dl>
<p>berechnet.
</p>
<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>
<ul><li><a href="Deviator" title="Deviator">Deviator</a></li>
<li><a href="Strecktensor" title="Strecktensor">Strecktensor</a></li>
<li><a href="Verzerrungstensor" title="Verzerrungstensor">Verzerrungstensor</a></li>
<li><a href="Formelsammlung_Tensoralgebra" title="Formelsammlung Tensoralgebra">Formelsammlung Tensoralgebra</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>H. Altenbach: <cite style="font-style:italic">Kontinuumsmechanik</cite>. Springer, 2012, ISBN 978-3-642-24118-5.<span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rfr_id=info:sid/de.wikipedia.org:Kugeltensor&amp;rft.au=H.+Altenbach&amp;rft.btitle=Kontinuumsmechanik&amp;rft.date=2012&amp;rft.genre=book&amp;rft.isbn=9783642241185&amp;rft.pub=Springer" style="display:none">&nbsp;</span></li>
<li>P. Haupt: <cite style="font-style:italic">Continuum Mechanics and Theory of Materials</cite>. Springer, 2000, ISBN 3-540-66114-X.<span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rfr_id=info:sid/de.wikipedia.org:Kugeltensor&amp;rft.au=P.+Haupt&amp;rft.btitle=Continuum+Mechanics+and+Theory+of+Materials&amp;rft.date=2000&amp;rft.genre=book&amp;rft.isbn=354066114X&amp;rft.pub=Springer" style="display:none">&nbsp;</span></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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